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Properties

Label 1-392-392.125-r1-0-0
Degree $1$
Conductor $392$
Sign $-0.958 + 0.284i$
Analytic cond. $42.1262$
Root an. cond. $42.1262$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (0.222 − 0.974i)17-s + 19-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (0.222 − 0.974i)29-s − 31-s + (0.900 − 0.433i)33-s + (0.222 − 0.974i)37-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (0.222 − 0.974i)17-s + 19-s + (−0.222 − 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.222 − 0.974i)27-s + (0.222 − 0.974i)29-s − 31-s + (0.900 − 0.433i)33-s + (0.222 − 0.974i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.958 + 0.284i$
Analytic conductor: \(42.1262\)
Root analytic conductor: \(42.1262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 392,\ (1:\ ),\ -0.958 + 0.284i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04759085291 - 0.3276120772i\)
\(L(\frac12)\) \(\approx\) \(-0.04759085291 - 0.3276120772i\)
\(L(1)\) \(\approx\) \(0.5957319119 - 0.1894704235i\)
\(L(1)\) \(\approx\) \(0.5957319119 - 0.1894704235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (0.222 - 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 - T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.24241712850381905440250784396, −23.71104781094019308537030033313, −23.111139014829512820762675171015, −22.028869166527564261782460044478, −21.50506325206469122738500079298, −20.44713111380265075166317609715, −19.31554845667724079774283545324, −18.52875245046116007920542953765, −17.78189479252828349484279642193, −16.46849269837608031338577706368, −16.07387889051506169275383071943, −15.206880788674351075959142377175, −14.16369785872460369574033623371, −12.948375235401242056847074477986, −11.90297785655173204882204711400, −11.18326619359898069369059098664, −10.58433129457245682473807802714, −9.3870380200715270088155830546, −8.204642635793480136271015544141, −7.17015758194447634910911403379, −6.14374518275147949831851936476, −5.20928863024324242014166331229, −3.968600425929895093925419512712, −3.26575949732017100650786078736, −1.28656228451433519188685762641, 0.13099568364788378807096559691, 1.09331744648682170937187877147, 2.70304649391606748726972713569, 4.205736190943151079466947880514, 5.08398748756207463354116984745, 6.017590314734240982487857704365, 7.48473789119262563329353869161, 7.708027645187886577381016694224, 9.18862069935463685770715399017, 10.42257765262297061907178665317, 11.2339899029477239250534441302, 12.18515158003421946079862656422, 12.730279719692875666260102245901, 13.74147747805341526614078466100, 15.15598112518728956511468133619, 16.02038966194889246676727746844, 16.50757451871050314952606376106, 17.88746617409631392814204048358, 18.227441621099861420386870853703, 19.33418766127826545929548768370, 20.32460762343915802656206498930, 20.98094347810305003668089794219, 22.54570924243716204159774433330, 22.82028803037390256052350350227, 23.64898300578218551364556638239

Graph of the $Z$-function along the critical line